author paulson Sun, 22 Jul 2018 13:29:41 +0200 changeset 68674 f4ac69fe4509 parent 68672 9247996782c9 (current diff) parent 68673 22d10f94811e (diff) child 68676 74cb08ff2e66
merged
```--- a/src/HOL/Algebra/QuotRing.thy	Sun Jul 22 13:00:38 2018 +0200
+++ b/src/HOL/Algebra/QuotRing.thy	Sun Jul 22 13:29:41 2018 +0200
@@ -16,61 +16,34 @@
where "rcoset_mult R I A B = (\<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b))"

-text \<open>@{const "rcoset_mult"} fulfils the properties required by
-  congruences\<close>
+text \<open>@{const "rcoset_mult"} fulfils the properties required by congruences\<close>
lemma (in ideal) rcoset_mult_add:
-    "x \<in> carrier R \<Longrightarrow> y \<in> carrier R \<Longrightarrow> [mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
-  apply rule
-  apply (rule, simp add: rcoset_mult_def, clarsimp)
-  defer 1
-  apply (rule, simp add: rcoset_mult_def)
-  defer 1
+  assumes "x \<in> carrier R" "y \<in> carrier R"
+  shows "[mod I:] (I +> x) \<Otimes> (I +> y) = I +> (x \<otimes> y)"
proof -
-  fix z x' y'
-  assume carr: "x \<in> carrier R" "y \<in> carrier R"
-    and x'rcos: "x' \<in> I +> x"
-    and y'rcos: "y' \<in> I +> y"
-    and zrcos: "z \<in> I +> x' \<otimes> y'"
-
-  from x'rcos have "\<exists>h\<in>I. x' = h \<oplus> x"
-    by (simp add: a_r_coset_def r_coset_def)
-  then obtain hx where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x"
-    by fast+
-
-  from y'rcos have "\<exists>h\<in>I. y' = h \<oplus> y"
-    by (simp add: a_r_coset_def r_coset_def)
-  then obtain hy where hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
-    by fast+
-
-  from zrcos have "\<exists>h\<in>I. z = h \<oplus> (x' \<otimes> y')"
-    by (simp add: a_r_coset_def r_coset_def)
-  then obtain hz where hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
-    by fast+
-
-  note carr = carr hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
-
-  from z have "z = hz \<oplus> (x' \<otimes> y')" .
-  also from x' y' have "\<dots> = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
-  also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
-  finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
-
-  from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
-    by (simp add: I_l_closed I_r_closed)
-
-  with z2 have "\<exists>h\<in>I. z = h \<oplus> x \<otimes> y" by fast
-  then show "z \<in> I +> x \<otimes> y" by (simp add: a_r_coset_def r_coset_def)
-next
-  fix z
-  assume xcarr: "x \<in> carrier R"
-    and ycarr: "y \<in> carrier R"
-    and zrcos: "z \<in> I +> x \<otimes> y"
-  from xcarr have xself: "x \<in> I +> x" by (intro a_rcos_self)
-  from ycarr have yself: "y \<in> I +> y" by (intro a_rcos_self)
-  show "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b"
-    using xself and yself and zrcos by fast
+  have 1: "z \<in> I +> x \<otimes> y"
+    if x'rcos: "x' \<in> I +> x" and y'rcos: "y' \<in> I +> y" and zrcos: "z \<in> I +> x' \<otimes> y'" for z x' y'
+  proof -
+    from that
+    obtain hx hy hz where hxI: "hx \<in> I" and x': "x' = hx \<oplus> x" and hyI: "hy \<in> I" and y': "y' = hy \<oplus> y"
+      and hzI: "hz \<in> I" and z: "z = hz \<oplus> (x' \<otimes> y')"
+      by (auto simp: a_r_coset_def r_coset_def)
+    note carr = assms hxI[THEN a_Hcarr] hyI[THEN a_Hcarr] hzI[THEN a_Hcarr]
+    from z  x' y' have "z = hz \<oplus> ((hx \<oplus> x) \<otimes> (hy \<oplus> y))" by simp
+    also from carr have "\<dots> = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" by algebra
+    finally have z2: "z = (hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy) \<oplus> x \<otimes> y" .
+    from hxI hyI hzI carr have "hz \<oplus> (hx \<otimes> (hy \<oplus> y)) \<oplus> x \<otimes> hy \<in> I"
+      by (simp add: I_l_closed I_r_closed)
+    with z2 show ?thesis
+      by (auto simp add: a_r_coset_def r_coset_def)
+  qed
+  have 2: "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b" if "z \<in> I +> x \<otimes> y" for z
+    using assms a_rcos_self that by blast
+  show ?thesis
+    unfolding rcoset_mult_def using assms
+    by (auto simp: intro!: 1 2)
qed

-
subsection \<open>Quotient Ring Definition\<close>

definition FactRing :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) ring"
@@ -79,59 +52,30 @@
\<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I,
one = (I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>), zero = I, add = set_add R\<rparr>"

+lemmas FactRing_simps = FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric]

subsection \<open>Factorization over General Ideals\<close>

text \<open>The quotient is a ring\<close>
lemma (in ideal) quotient_is_ring: "ring (R Quot I)"
-apply (rule ringI)
-   \<comment> \<open>abelian group\<close>
-   apply (rule comm_group_abelian_groupI)
-   apply (simp add: FactRing_def)
-   apply (rule a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
-  \<comment> \<open>mult monoid\<close>
-  apply (rule monoidI)
-      apply (simp_all add: FactRing_def A_RCOSETS_def RCOSETS_def
-             a_r_coset_def[symmetric])
-      \<comment> \<open>mult closed\<close>
-      apply (clarify)
-     \<comment> \<open>mult \<open>one_closed\<close>\<close>
-     apply force
-    \<comment> \<open>mult assoc\<close>
-    apply clarify
-   \<comment> \<open>mult one\<close>
-   apply clarify
-  apply clarify
- \<comment> \<open>distr\<close>
- apply clarify
- apply (simp add: rcoset_mult_add a_rcos_sum l_distr)
-apply clarify
-done
+proof (rule ringI)
+  show "abelian_group (R Quot I)"
+    apply (rule comm_group_abelian_groupI)
+    apply (simp add: FactRing_def a_factorgroup_is_comm_group[unfolded A_FactGroup_def'])
+    done
+  show "Group.monoid (R Quot I)"
+    by (rule monoidI)
+      (auto simp add: FactRing_simps rcoset_mult_add m_assoc)
+qed (auto simp: FactRing_simps rcoset_mult_add a_rcos_sum l_distr r_distr)

text \<open>This is a ring homomorphism\<close>

lemma (in ideal) rcos_ring_hom: "((+>) I) \<in> ring_hom R (R Quot I)"
-apply (rule ring_hom_memI)
-   apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
- apply (simp add: FactRing_def a_rcos_sum)
-apply (simp add: FactRing_def)
-done
+  by (simp add: ring_hom_memI FactRing_def a_rcosetsI[OF a_subset] rcoset_mult_add a_rcos_sum)

lemma (in ideal) rcos_ring_hom_ring: "ring_hom_ring R (R Quot I) ((+>) I)"
-apply (rule ring_hom_ringI)
-     apply (rule is_ring, rule quotient_is_ring)
-   apply (simp add: FactRing_def a_rcosetsI[OF a_subset])
- apply (simp add: FactRing_def a_rcos_sum)
-apply (simp add: FactRing_def)
-done
+  by (simp add: local.ring_axioms quotient_is_ring rcos_ring_hom ring_hom_ringI2)

text \<open>The quotient of a cring is also commutative\<close>
lemma (in ideal) quotient_is_cring:
@@ -140,12 +84,9 @@
proof -
interpret cring R by fact
show ?thesis
-    apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro)
-      apply (rule quotient_is_ring)
+    apply (intro cring.intro comm_monoid.intro comm_monoid_axioms.intro quotient_is_ring)
apply (rule ring.axioms[OF quotient_is_ring])
-    apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric])
-    apply clarify
+    apply (auto simp add: FactRing_simps rcoset_mult_add m_comm)
done
qed

@@ -169,37 +110,19 @@

text \<open>The quotient ring generated by a prime ideal is a domain\<close>
lemma (in primeideal) quotient_is_domain: "domain (R Quot I)"
-  apply (rule domain.intro)
-   apply (rule quotient_is_cring, rule is_cring)
-  apply (rule domain_axioms.intro)
-   apply (simp add: FactRing_def) defer 1
-    apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarify)
-    apply (simp add: rcoset_mult_add) defer 1
-proof (rule ccontr, clarsimp)
-  assume "I +> \<one> = I"
-  then have "\<one> \<in> I" by (simp only: a_coset_join1 one_closed a_subgroup)
-  then have "carrier R \<subseteq> I" by (subst one_imp_carrier, simp, fast)
-  with a_subset have "I = carrier R" by fast
-  with I_notcarr show False by fast
-next
-  fix x y
-  assume carr: "x \<in> carrier R" "y \<in> carrier R"
+proof -
+  have 1: "I +> \<one> = I \<Longrightarrow> False"
+    using I_notcarr a_rcos_self one_imp_carrier by blast
+  have 2: "I +> x = I"
+    if  carr: "x \<in> carrier R" "y \<in> carrier R"
and a: "I +> x \<otimes> y = I"
-    and b: "I +> y \<noteq> I"
-
-  have ynI: "y \<notin> I"
-  proof (rule ccontr, simp)
-    assume "y \<in> I"
-    then have "I +> y = I" by (rule a_rcos_const)
-    with b show False by simp
-  qed
-
-  from carr have "x \<otimes> y \<in> I +> x \<otimes> y" by (simp add: a_rcos_self)
-  then have xyI: "x \<otimes> y \<in> I" by (simp add: a)
-
-  from xyI and carr have xI: "x \<in> I \<or> y \<in> I" by (simp add: I_prime)
-  with ynI have "x \<in> I" by fast
-  then show "I +> x = I" by (rule a_rcos_const)
+    and b: "I +> y \<noteq> I" for x y
+    by (metis I_prime a a_rcos_const a_rcos_self b m_closed that)
+  show ?thesis
+    apply (intro domain.intro quotient_is_cring is_cring domain_axioms.intro)
+     apply (metis "1" FactRing_def monoid.simps(2) ring.simps(1))
+    apply (simp add: FactRing_simps)
+    by (metis "2" rcoset_mult_add)
qed

text \<open>Generating right cosets of a prime ideal is a homomorphism
@@ -210,88 +133,58 @@

subsection \<open>Factorization over Maximal Ideals\<close>

-text \<open>In a commutative ring, the quotient ring over a maximal ideal
-        is a field.
-        The proof follows ``W. Adkins, S. Weintraub: Algebra --
-        An Approach via Module Theory''\<close>
-lemma (in maximalideal) quotient_is_field:
+text \<open>In a commutative ring, the quotient ring over a maximal ideal is a field.
+        The proof follows ``W. Adkins, S. Weintraub: Algebra -- An Approach via Module Theory''\<close>
+proposition (in maximalideal) quotient_is_field:
assumes "cring R"
shows "field (R Quot I)"
proof -
interpret cring R by fact
-  show ?thesis
-    apply (intro cring.cring_fieldI2)
-      apply (rule quotient_is_cring, rule is_cring)
-     defer 1
-     apply (simp add: FactRing_def A_RCOSETS_defs a_r_coset_def[symmetric], clarsimp)
-     apply (simp add: rcoset_mult_add) defer 1
-  proof (rule ccontr, simp)
-    \<comment> \<open>Quotient is not empty\<close>
+  have 1: "\<zero>\<^bsub>R Quot I\<^esub> \<noteq> \<one>\<^bsub>R Quot I\<^esub>"  \<comment> \<open>Quotient is not empty\<close>
+  proof
assume "\<zero>\<^bsub>R Quot I\<^esub> = \<one>\<^bsub>R Quot I\<^esub>"
then have II1: "I = I +> \<one>" by (simp add: FactRing_def)
-    from a_rcos_self[OF one_closed] have "\<one> \<in> I"
-      by (simp add: II1[symmetric])
-    then have "I = carrier R" by (rule one_imp_carrier)
+    then have "I = carrier R"
+      using a_rcos_self one_imp_carrier by blast
with I_notcarr show False by simp
-  next
+  qed
+  have 2: "\<exists>y\<in>carrier R. I +> a \<otimes> y = I +> \<one>" if IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R" for a
\<comment> \<open>Existence of Inverse\<close>
-    fix a
-    assume IanI: "I +> a \<noteq> I" and acarr: "a \<in> carrier R"
-
+  proof -
\<comment> \<open>Helper ideal \<open>J\<close>\<close>
define J :: "'a set" where "J = (carrier R #> a) <+> I"
have idealJ: "ideal J R"
-      apply (unfold J_def, rule add_ideals)
-       apply (simp only: cgenideal_eq_rcos[symmetric], rule cgenideal_ideal, rule acarr)
-      apply (rule is_ideal)
-      done
-
-    \<comment> \<open>Showing @{term "J"} not smaller than @{term "I"}\<close>
+      using J_def acarr add_ideals cgenideal_eq_rcos cgenideal_ideal is_ideal by auto
have IinJ: "I \<subseteq> J"
-    proof (rule, simp add: J_def r_coset_def set_add_defs)
+    proof (clarsimp simp: J_def r_coset_def set_add_defs)
fix x
assume xI: "x \<in> I"
-      have Zcarr: "\<zero> \<in> carrier R" by fast
-      from xI[THEN a_Hcarr] acarr
-      have "x = \<zero> \<otimes> a \<oplus> x" by algebra
-      with Zcarr and xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
-    qed
-
-    \<comment> \<open>Showing @{term "J \<noteq> I"}\<close>
-    have anI: "a \<notin> I"
-    proof (rule ccontr, simp)
-      assume "a \<in> I"
-      then have "I +> a = I" by (rule a_rcos_const)
-      with IanI show False by simp
+      have "x = \<zero> \<otimes> a \<oplus> x"
+        by (simp add: acarr xI)
+      with xI show "\<exists>xa\<in>carrier R. \<exists>k\<in>I. x = xa \<otimes> a \<oplus> k" by fast
qed
-
-    have aJ: "a \<in> J"
-    proof (simp add: J_def r_coset_def set_add_defs)
-      from acarr
-      have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
-      show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
+    have JnI: "J \<noteq> I"
+    proof -
+      have "a \<notin> I"
+        using IanI a_rcos_const by blast
+      moreover have "a \<in> J"
+      proof (simp add: J_def r_coset_def set_add_defs)
+        from acarr
+        have "a = \<one> \<otimes> a \<oplus> \<zero>" by algebra
+        show "\<exists>x\<in>carrier R. \<exists>k\<in>I. a = x \<otimes> a \<oplus> k" by fast
+      qed
+      ultimately show ?thesis by blast
qed
-
-    from aJ and anI have JnI: "J \<noteq> I" by fast
-
-    \<comment> \<open>Deducing @{term "J = carrier R"} because @{term "I"} is maximal\<close>
-    from idealJ and IinJ have "J = I \<or> J = carrier R"
-    proof (rule I_maximal, unfold J_def)
-      have "carrier R #> a \<subseteq> carrier R"
-        using subset_refl acarr by (rule r_coset_subset_G)
-      then show "carrier R #> a <+> I \<subseteq> carrier R"
-        using a_subset by (rule set_add_closed)
-    qed
-
-    with JnI have Jcarr: "J = carrier R" by simp
+    then have Jcarr: "J = carrier R"
+      using I_maximal IinJ additive_subgroup.a_subset idealJ ideal_def by blast

\<comment> \<open>Calculating an inverse for @{term "a"}\<close>
from one_closed[folded Jcarr]
-    have "\<exists>r\<in>carrier R. \<exists>i\<in>I. \<one> = r \<otimes> a \<oplus> i"
-      by (simp add: J_def r_coset_def set_add_defs)
-    then obtain r i where rcarr: "r \<in> carrier R"
-      and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i" by fast
+    obtain r i where rcarr: "r \<in> carrier R"
+      and iI: "i \<in> I" and one: "\<one> = r \<otimes> a \<oplus> i"
+      by (auto simp add: J_def r_coset_def set_add_defs)
+
from one and rcarr and acarr and iI[THEN a_Hcarr]
have rai1: "a \<otimes> r = \<ominus>i \<oplus> \<one>" by algebra

@@ -305,6 +198,10 @@
from rcarr and this[symmetric]
show "\<exists>r\<in>carrier R. I +> a \<otimes> r = I +> \<one>" by fast
qed
+  show ?thesis
+    apply (intro cring.cring_fieldI2 quotient_is_cring is_cring 1)
+     apply (clarsimp simp add: FactRing_simps rcoset_mult_add 2)
+    done
qed

@@ -916,7 +813,7 @@
proof -
fix X assume "X \<in> carrier (R Quot (a_kernel R S h))"
thus "\<exists>x \<in> carrier R. X = (a_kernel R S h) +> x"
-    unfolding FactRing_def RCOSETS_def A_RCOSETS_def by (simp add: a_r_coset_def)
+    unfolding FactRing_simps by (simp add: a_r_coset_def)
qed

lemma (in ring_hom_ring) the_elem_wf:
@@ -989,7 +886,7 @@
by (metis image_iff)
hence "the_elem (h ` ((a_kernel R S h) +> x)) = y" by simp
moreover have "(a_kernel R S h) +> x \<in> carrier (R Quot (a_kernel R S h))"
-     unfolding FactRing_def RCOSETS_def A_RCOSETS_def by (auto simp add: x a_r_coset_def)
+     unfolding FactRing_simps by (auto simp add: x a_r_coset_def)
ultimately show "y \<in> (\<lambda>X. (the_elem (h ` X))) ` carrier (R Quot (a_kernel R S h))" by blast
qed
qed```
```--- a/src/HOL/Algebra/Ring.thy	Sun Jul 22 13:00:38 2018 +0200
+++ b/src/HOL/Algebra/Ring.thy	Sun Jul 22 13:29:41 2018 +0200
@@ -332,7 +332,7 @@
assumes "cring R"
shows "comm_monoid R"
and "\<And>x y z. \<lbrakk> x \<in> carrier R; y \<in> carrier R; z \<in> carrier R \<rbrakk> \<Longrightarrow> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
-  using assms cring_def apply auto by (simp add: assms cring.axioms(1) ringE(3))
+  using assms cring_def by auto (simp add: assms cring.axioms(1) ringE(3))

lemma (in cring) is_cring:
"cring R" by (rule cring_axioms)
@@ -481,10 +481,7 @@
qed

lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
-  apply rule
-   apply (erule one_zeroI)
-  apply (erule one_zeroD)
-  done
+  using one_zeroD by blast

lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
by (simp add: carrier_one_zero)
@@ -687,19 +684,22 @@
text \<open>Another variant to show that something is a field\<close>
lemma (in cring) cring_fieldI2:
assumes notzero: "\<zero> \<noteq> \<one>"
-  and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
+    and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
shows "field R"
-  apply (rule cring_fieldI, simp add: Units_def)
-  apply (rule, clarsimp)
-  apply (simp add: notzero)
-proof (clarsimp)
-  fix x
-  assume xcarr: "x \<in> carrier R"
-    and "x \<noteq> \<zero>"
-  then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
-  then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast
-  from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
-  with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
+proof -
+  have *: "carrier R - {\<zero>} \<subseteq> {y \<in> carrier R. \<exists>x\<in>carrier R. x \<otimes> y = \<one> \<and> y \<otimes> x = \<one>}"
+  proof (clarsimp)
+    fix x
+    assume xcarr: "x \<in> carrier R" and "x \<noteq> \<zero>"
+    obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>"
+      using \<open>x \<noteq> \<zero>\<close> invex xcarr by blast
+    with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>"
+      using m_comm xcarr by fastforce
+  qed
+  show ?thesis
+    apply (rule cring_fieldI, simp add: Units_def)
+    using *
+    using group_l_invI notzero set_diff_eq by auto
qed

@@ -859,57 +859,36 @@

subsection\<open>Jeremy Avigad's @{text"More_Ring"} material\<close>

-lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub> \<Longrightarrow> \<forall>x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}. x \<in> Units R \<Longrightarrow> field R"
-  apply (unfold_locales)
-    apply (use cring_axioms in auto)
-   apply (rule trans)
-    apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
-     apply assumption
-    apply (subst m_assoc)
-       apply auto
-  apply (unfold Units_def)
-  apply auto
-  done
+lemma (in cring) field_intro2:
+  assumes "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>" and un: "\<And>x. x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>} \<Longrightarrow> x \<in> Units R"
+  shows "field R"
+proof unfold_locales
+  show "\<one> \<noteq> \<zero>" using assms by auto
+  show "\<lbrakk>a \<otimes> b = \<zero>; a \<in> carrier R;
+            b \<in> carrier R\<rbrakk>
+           \<Longrightarrow> a = \<zero> \<or> b = \<zero>" for a b
+    by (metis un Units_l_cancel insert_Diff_single insert_iff r_null zero_closed)
+qed (use assms in \<open>auto simp: Units_def\<close>)

lemma (in monoid) inv_char:
-  "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
-  apply (subgoal_tac "x \<in> Units G")
-   apply (subgoal_tac "y = inv x \<otimes> \<one>")
-    apply simp
-   apply (erule subst)
-   apply (subst m_assoc [symmetric])
-      apply auto
-  apply (unfold Units_def)
-  apply auto
-  done
+  assumes "x \<in> carrier G" "y \<in> carrier G" "x \<otimes> y = \<one>" "y \<otimes> x = \<one>"
+  shows "inv x = y"
+  using assms inv_unique' by auto

lemma (in comm_monoid) comm_inv_char: "x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
by (simp add: inv_char m_comm)

lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
-  apply (rule inv_char)
-     apply (auto simp add: l_minus r_minus)
-  done
+  by (simp add: inv_char local.ring_axioms ring.r_minus)

lemma (in monoid) inv_eq_imp_eq: "x \<in> Units G \<Longrightarrow> y \<in> Units G \<Longrightarrow> inv x = inv y \<Longrightarrow> x = y"
-  apply (subgoal_tac "inv (inv x) = inv (inv y)")
-   apply (subst (asm) Units_inv_inv)+
-    apply auto
-  done
+  by (metis Units_inv_inv)

lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> \<in> Units R"
-  apply (unfold Units_def)
-  apply auto
-  apply (rule_tac x = "\<ominus> \<one>" in bexI)
-   apply auto
-  apply (simp add: l_minus r_minus)
-  done
+  by (simp add: Units_def) (metis add.l_inv_ex local.minus_minus minus_equality one_closed r_minus r_one)

lemma (in ring) inv_eq_neg_one_eq: "x \<in> Units R \<Longrightarrow> inv x = \<ominus> \<one> \<longleftrightarrow> x = \<ominus> \<one>"
-  apply auto
-  apply (subst Units_inv_inv [symmetric])
-   apply auto
-  done
+  by (metis Units_inv_inv inv_neg_one)

lemma (in monoid) inv_eq_one_eq: "x \<in> Units G \<Longrightarrow> inv x = \<one> \<longleftrightarrow> x = \<one>"
by (metis Units_inv_inv inv_one)```